Lens Space Surgeries and the Bleiler-Litherland Conjecture
Abstract
We prove the Bleiler-Litherland conjecture: every lens space obtained by a nontrivial Dehn surgery on a hyperbolic knot in S3 has order at least 18. The key ingredient is a spectral obstruction: if a hyperbolic knot K of genus g admits a lens space surgery with slope ±(4g−2), then ΔK(t) has a real root outside the unit circle. The proof combines Floer-theoretic restrictions on lens space surgeries, Gabai's degeneracy-slope bound and Gabai-Oertel's persistence theorem for essential laminations, Ni's fixed-point theorem for monodromy, and a mod-2 orientability criterion, together with the relation between homological monodromy and the Alexander polynomial. As a further application of this spectral obstruction, we obtain characterizing-slope results for torus knots.
Related
- Lens space surgeries on knots of small genus
- Lens spaces that cannot be obtained by two integral surgeries
- L-space surgeries on (1,1)-knots in S1× S2
- The d-invariant of surgeries on $2$-component L-space links
- Supercritical magnetic ray transforms in hyperbolic space: the complexification method from the transport-twistor lens
BibTeX
@misc{guo2026lens,
title = {{Lens Space Surgeries and the Bleiler-Litherland Conjecture}},
author = {Qilong Guo},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.31757},
eprint = {2609.31757},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.31757},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Lens Space Surgeries and the Bleiler-Litherland Conjecture
AU - Qilong Guo
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.31757
UR - https://arxiv.org/abs/2609.31757
AB - We prove the Bleiler-Litherland conjecture: every lens space obtained by a nontrivial Dehn surgery on a hyperbolic knot in S^3 has order at least 18. The key ingredient is a spectral obstruction: if a hyperbolic knot K of genus g admits a lens space surgery with slope ±(4g−2), then Δ_K(t) has a real root outside the unit circle. The proof combines Floer-theoretic restrictions on lens space surgeries, Gabai's degeneracy-slope bound and Gabai-Oertel's persistence theorem for essential laminations, Ni's fixed-point theorem for monodromy, and a mod-2 orientability criterion, together with the relation between homological monodromy and the Alexander polynomial. As a further application of this spectral obstruction, we obtain characterizing-slope results for torus knots.
ER -